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     U   N   E  S  C  O   –    E  O   L  S  S  S  A   M   P   L   E   C   H  A   P   T   E   R  S CONTROL SYSTEMS, ROBOTICS AND AUTOMATION – Vol. II -  Controller Design in Time-Domain  - Unbehauen H. ©Encyclopedia of Life Support Systems  (EOLSS) CONTROLLER DESIGN IN TIME-DOMAIN Unbehauen H. Control Engineering Division,    Department of Electrical Engineering and Information Sciences, Ruhr University Bochum, Germany   Keywords:  Tracking control, Disturbance rejection, Performance specifications, Integral criteria, Optimal controller setting, Tuning rules for standard controllers, Empirical design, Standard polynomials. Contents 1. Problem formulation 2. Time-domain performance specifications 2.1. Transient Performance 2.2. Integral Criteria 2.3. Calculation of the ISE-Performance Index 3. Optimal controller settings subject to the ISE-criterion 3.1. Example 3.2. Optimal Settings for Combinations of n PT -Plants and Standard Controllers of PID Type 4. Empirical procedures 4.1. Tuning Rules for Standard Controllers 4.1.1. Ziegler-Nicols Tuning Rules 4.1.2. Some Other Useful Tuning Rules 4.2. Empirical Design by Computer Simulation 5. Mixed time- and frequency-domain design by standard polynomials 6. Concluding remarks Glossary Bibliography Biographical Sketch Summary This article presents an introduction to the classical design methods for linear continuous time-invariant single input/single output control systems in the time-domain. The design is based on finding the “best” possible controller with respect to selected time-domain  performance specifications. For the dynamic behavior of the closed-loop control system,  performance specifications are defined for the input step responses of the reference signal and disturbance. These transient performance specifications are natural and are used to formulate the desired closed-loop behavior of the control system. However, these specifications are more appropriate for evaluating the result of a control system design, whereas the design is usually based on minimizing specific integral performance indices using various functions of the error between the reference input and the controlled plant output. Especially in the case of a fixed controller structure, these integral criteria provide optimal controller settings. The solution of this optimization problem can be obtained by numerical or analytical approaches. In the time-domain design, empirical procedures, such as tuning rules for standard controllers or design by computer simulation play an     U   N   E  S  C  O   –    E  O   L  S  S  S  A   M   P   L   E   C   H  A   P   T   E   R  S CONTROL SYSTEMS, ROBOTICS AND AUTOMATION – Vol. II -  Controller Design in Time-Domain  - Unbehauen H. ©Encyclopedia of Life Support Systems  (EOLSS) important role. Time-domain specifications can also be used to select standard-polynomials, such as the characteristic polynomial for the desired closed-loop transient behavior. This leads to a mixed time- and frequency-domain design, where the solution provides the structure and parameters of the controller as a result of the selected time-domain performance specification. 1. Problem Formulation The design of a control system may lead to different solutions to meet explicit design goals, but also implicit engineering goals such as economical considerations, complexity and reliability. The design procedure depends on whether the nominal plant transfer function P () Gs  is known or not. In any case, the “best” possible controller or compensator transfer function C () Gs  has to  be designed or selected and tuned such that the desired performance specifications are met. In general the designed closed-loop system, considered in Figure 1, should at least fulfil the following conditions: 1)   The closed-loop system has to be stable. 2)   Disturbances () dt   should have only a minimal influence on the controlled variable ()  yt  . 3)   The controlled variable ()  yt   must be able to track the reference signal () rt   as fast and as accurately as possible. 4)   The closed-loop system should not be too sensitive to parameter changes of the plant. In order to fulfil conditions 2) and 3) the closed-loop transfer function for tracking control in the ideal case should be, assuming unity feedback 0R 0 ()()()1()1() GsYsGs RsGs = = =+ , (1) where 0CP ()()() GsGsGs =  is the open-loop transfer function, and the corresponding ideal transfer function for the closed-loop in the case of disturbance rejection should be D0 ()1()0()1() YsGs DsGs = = =+  (2) Theoretically, Eqs. (1) and (2) can only be satisfied if 0 ()1 Gss >> ∀ , which will be the case for a large value of the gain factor 0 1 K   >>  of 0 () Gs , where 0 K   is the gain factor of 0 () Gs . It should be noted that in this article only unity feedback is considered. The addition of a feedback controller can enhance stability and design flexibility.     U   N   E  S  C  O   –    E  O   L  S  S  S  A   M   P   L   E   C   H  A   P   T   E   R  S CONTROL SYSTEMS, ROBOTICS AND AUTOMATION – Vol. II -  Controller Design in Time-Domain  - Unbehauen H. ©Encyclopedia of Life Support Systems  (EOLSS) Figure 1. Block diagram of a standard linear closed-loop control system However, both conditions, Eqs. (1) and (2), cannot be satisfied strictly, due to physical limitations especially concerning the controller gain and the magnitude of the manipulating signal. Furthermore, increasing 0 K   too much would lead in most cases to stability problems. In practice, the design engineer has to make a compromise between the desired behavior and the technical limitations. This procedure needs a lot of experience, and engineering judgement, as well as intuition. Thus, it is understandable that for the design of control systems either in the frequency- or time-domain many different approaches are available and provide different solutions. Each solution is optimal with respect to the selected measure of performance. In this article only some classical design methods in the time-domain are considered. The design of state feedback controllers is, therefore, discussed separately (see  Design of State Space Controllers for SISO Systems ). 2. Time-Domain Performance Specifications The starting point for the design of a feedback-control system is to have a good plant model described either in the form of a differential equation or a transfer function P () Gs . Once the plant model is given, the next step is to design an overall system, as shown in Figure 1, that meets the desired design specifications. It is important to note that different applications may require different specifications. Generally, the performance of feedback-control systems includes two tasks: steady-state  performance, which specifies accuracy when all the transients are decayed (see Closed-loop Behvior  ), and transient performance, which specifies the speed of response as discussed below. 2.1. Transient Performance The transient performance is usually defined for a step reference or step disturbance input response as shown in Figure 2. The specifications indicated in Figure 2 are natural. In the case of reference tracking (see Figure 2a) these specifications are as follows:     U   N   E  S  C  O   –    E  O   L  S  S  S  A   M   P   L   E   C   H  A   P   T   E   R  S CONTROL SYSTEMS, ROBOTICS AND AUTOMATION – Vol. II -  Controller Design in Time-Domain  - Unbehauen H. ©Encyclopedia of Life Support Systems  (EOLSS) Figure 2. Step responses for (a) reference input and (b) disturbance input including main  parameters of transient performance Peak overshoot    max e : This term is defined as the maximum value of the response at time max t   in relation to its desired final value. It can be considered to be a measure of the relative stability of the system. It increases as the damping ratio decreases.  Rise time   a T  : Is defined often as the time required for a response to go from 10 % to 90 % of its desired final value, or as the time interval given by the intersection points of the inflexion tangent with the 0 % and 100 % lines.  Delay time   u T  : This is the time between the excitation and the intersection point of the inflexion tangent of the response with the 0 % line. Settling time   t  ε  : This term is the time after which the response remains within a band of % ε  ± about the desired final value, where ε   is selected between 2 % and 5 %.  Reaching time   an t  : This is the time at which the response reaches for the first time the desired final value, where anua tTT  ≈ + . Similarly, the case of disturbance rejection (see Figure 2b) can be characterized by introducing the peak overshoot and settling time. Whereas max e  and t  ε   depend upon the damping ratio, the other values a T  , max t   and an t   represent a measure for the speed of the transient behavior.
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